There has been tremendous amount of work towards understanding the non-Markovian behaviour in single and many qubits coupled to a bosonic bath[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. In most of these cases, Born approximation is used which means that the joint initial state of the system and bath
are assumed to be uncorrelated. However this assumption is often violated when there exist strong interaction between system and the bath [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}. In particular, quantum state preparation can lead to strong system-bath correlations thus effecting the subsequent dynamics. In this regard, various works have critcally examined the initial system-bath correlations in spin-Boson model[18]}, superconducting qubits [19]}, quantum dots [20]} etc. For example, Mozorov et. al. [12]}, [18]}, has considered an exactly solvable dephasing model of a single qubit interacting a bosonic in presence of initial system-bath correlations, while in [23]} has considered Jaynes-Cunnings model and [24]} has studied in detail dynamics in presence of spin bath. These models however, consider either the single qubit or many coupled with individual baths. Furthermore, the distance of separation between the qubits in presence of initial correlations is not taken into account in these works. Our main objective in this work is to study the effect of distance dependent interaction between qubits and the collective bath in presence of initial system-bath correlations. Such kind of settings can be obtained using cold atom impurities immersed in a Bose-Einestien condensate (BEC) [25]}. The positioning of immersing of cold atoms in the BEC would yeild a distance dependent interaction with non-trivial bath spectral density. Thus the present work would be mainly important for quantum information processings using cold atoms.
